by Passive Arrays. MAT1 WAX AND THOMAS KAILATH, FELLOW, IEEE. Abstract-The maximum likelihood (MIL) estimator of the location of multiple sources and ...
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IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL
However, the nth component of
(C2) gives
[ 101 J. Risannen, “Algorithms for triangular decomposition of block
_v,(n, n) =_vm(n,n - 1)(1 - trn-l(n - 1, n - 1)) and substituting this in (C3)gives follows from (C1) and (15d).
PROCESSING, VOL. ASSP-31, NO. 5 , OCTOBER 1983
(37a). Equation (37b)
[ 111 [ 121
REFERENCES [ 11 H. Akaike, “Markovian representation of stochastic processes and its application to the analysis of autoregressive movingaverage processes,”Ann. Znst. Stat. Math., vol. 20, p. 363, 1974. [21 J. Cadzow, estimation-A new
method,” ZEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-28, P. 524, 1980. [3] B. Friedlander, ‘“Lattice implementationof some recursive parameter estimation algorithms,” in Proc. 6th IFAC Symp. Ident. Syst. Parameter Estimation, p. 481, 1982. 141G. Carayannis, N. Kalouptsidis, and D. Manolakis, “Efficient algorithm for determination of A.R. part of ARMA model,” Electron. Lett., pp. 656-657, 1980. [5] S. M. Kay,’ “A new ARMA spectralestimator,” ZEEE Trans. Acoust., Speech, SignalProcessing, vol ASSP-28, p. 585, 1980. [6] D. T. Lee, B. Friedlander, and M. Morf, “Recursive ladder algorithms for ARMA modelling,” in Proc. 19th ZEEE Con5 Decision Contr., pp. 1225-1231, Dec. 1980. [7] D. T. Lee, M. Morf, and B. Friedlander, “Recursive least squares ladder estimation algorithms,” ZEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 627-641, 1981. [8] L. Ljung, M. Morf, and D. Falconer, “Fast calculation of gain matrices for recursive estimation schemes,”Znt. J. Contr.,vol. 27, pp. 1-19, 1978. [9] J. Makhoul, “Stable and efficient lattice methods for linear prediction,” IEEE Trans. Acoust.,Speech, Signal Processing, vol. ASSP-25, pp. 423-428, 1977.
[13] P. [ 141
Hankel and Toeplitz matrices with application to factoring positive matrix polynomials,” Math. Cornput., vol. 17, pp. 147-154, 1973. C. Samson, “A unified treatment of fast algorithms for identification,”Znt. J. Contr., vol. 35, pp. 909-934, 1982. E. H. Sartorius and M. J. Shensa, “Recursive lattice Titers-A brief overview,” in Proc. 19th ZEEE Con5 Decision Contr., pp. 955959, Dec. 1980. M. J. Shensa, “Recursive least squares lattice algorithms-Ageometrical approach,” ZEEE Trans. Automat. Contr., vol.AC-26, pp. 695-702,1981. Whittle, “On the fitting of multivariable autoregressions and the approximate canonical factorization of the spectral density matrix,” Biometrika, vol. 50, p. 129, 1963.
. Victor Solo (”82)was
born in Mareeba, AusHereceived the tralia, on January18,1951. B.Sc. degree in mathematics from the University of Queensland, the B.Sc. degree in statistics (1st class honors) and the.B.E. degree in mechanical engineering (1st class honors) from the University of New South Wales, and the Ph.D. ’degree in statistics from Australian National University in 1971, 1973, 1974, and 1979, respectively. For threeperiods(t0taling two years) between 1974 and 1979, he worked for the CSIRO, Division of Mathematics and Statistics, as a Consulting Statistician. From 1979 to 1980he was an Assistant Research Scientist at the Mathematics Research Center, University of Wisconsin, Madison. Since the middle of 1980 he has been an Assistant Professor in the Department of Statistics at Harvard Univesity, Cambridge, MA. His research interests are in time series, econometrics, signal processing, linear systems, and control.
Optimum Localization of Multiple Sources by Passive Arrays MAT1 WAX
AND
THOMAS KAILATH, FELLOW,
Abstract-The maximum likelihood (MIL) estimator of the location of multiple sources and the corresponding Cramer-Rao lower bound on the error covariance matrix are derived. The derivation is carried out for the general case of correlated sources so that multipath propagation is included as a special case. It is shown that theML processor consists of a bank of beam-formers, each focused to a different source, followed by a variable matrix-filter that is controlled by the assumed location of the sources. In the special case of uncorrelated sources and very low signal-to-noise ratio this Manuscript received July 29, 1982; revised November 30, 1982, and March 3, 1983. This work was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command under Contract AF49-620-70-C-0058, and by the U.S. Army Research Office under Contract DAAG29-79-C-0215. The authors are with the Information Systems Laboratory, Stanford University, Stanford, CA 94305.
IEEE
processor reduces to an aggregate of ML processors for a single source with each processor matched to a different source. Iterative algorithms for the actual computation of the ML estimator are also presented.
I.
T
INTRODUCTION
HE problem of estimating the location of radiating sources frommeasurementsprovided by an arrayofsensors arises frequently in radar, sonar, and seismology. The signals received by the sensors consist, in the simplest case, of scaled and delayed replicas of the wavefronts radiated bythe sources. In a morecomplicated scenario, multiplepropagationpaths from the sourcesto the sensors may exist. The problem of localizing a singre source by a passive array
0096-3518/83/1000-1210$01.00 0 1983 IEEE
WAX AND KAILATH: OPTIMUM LOCALIZATION OF MULTIPLE SOURCES
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has been extensively investigated in the last two decades (see, our derivation is in the frequency domain, which allows us to e.g., [ 6 ] ) . The conventional approach to this problem is based give adirectphysicallyinterpretablerealizationofthereceived structure. It is shown that this processor is a natural on a two-step procedure. First, the time-differences-of-arrival (TDOA’s) of the propagating signal to the different sensors are extension of the ML processor for a single source derived by estimated, using aformof generalized correlator (see, e.g., Hahn and Tretter [7]. The Cramer-Rao lower bound on the also derived. The [9] ). Then the corresponding lines-of-position arederived and error covariance matrix for this problem is expressions obtained are complicated and unenlightening for “optimally”intersected to obtainthe sourcelocation. This method, being a two-step procedure, is suboptimal. The op- the general case. Nevertheless, they are amenable to numerical serveas astandardreference forany timal ML processor for the single source problem was derived evaluationandcan estimator. The computational issues of the ML estimator are by Bangs and Schultheiss [2] in terms of the estimate of the range andbearingofthesource.Anotherformofthe ML addressed and iterative algorithms are outlined and discussed. processor was described by Hahn and Tretter [7]. Their pro11. MODEL FORMULATION cessor was actually designed to estimatethe TDOA’s of the Assume that we have m sensors and d sources distributed in propagating signal, but it can be easily modified to estimate directly the coordinates of the source. Both versions of the space, as shown in Fig. 1. Each source is assumed to emit a signal that propagates radially with speed c. The sources may ML processorrequire a prioriknowledgeofthespectral densities of the signal and the noises. Bangs and Schultheiss be correlated, as will clearly be the case of multipath propagareceived signal at the ith sensor by [ 2 ] also derived the Cramer-Rao lower bound on the accuracy tion exists. Denoting the p i ( * ) , we can write of the estimator of the source location. The performanceofthe generalized correlator is severely d 4 ~ < . i .< m 1 ri(t) = aik sk(t - ~ i k )+ n&t) degradedinthepresence of multiple sources ormultipath k= 1 T/2< t < T/2 propagationthusrequiringadifferentscheme for TDOA estimation in these cases. An interesting solution, outlined by (11 Morf et al. [lo] ,is to model the received signals by parametric where (ARMA) models. Using system identification techniques, the sk(.) = the signal radiated by the kth source, TDOA estimates can then be extractedfromthe received T~~ = the propagation time from the kth source to the ith signals. T h s idea was furtherelaboratedand developed by sensor O i k = rik/c where rik is the spatial separation), PoratandFriedlander[13]and Nehorai and Morf [ 111. aik = the propagation attenuation from the kth source to The shortcomingofthesemethods, as of all parametric the ith sensor (proportional to l/rik), methods, is theirsensitivity to theformof the assumed n i(.) = the additive noise at the ithsensor. model. Another approach to the problem of localizing sources by We assume that the following hold. 1) The sources {sk(*)} are wide-band stationary zero-mean a passive array,which can copewithmultiplesourcesand multipath propagation as well, was recently presented by Wax jointly Gaussian processes, uncorrelatedwiththeadditive noises. et al. [I81 . This method is nonparametric and hence robust, 2) The additivenoises {ni(*)} are wide-bandstationary but even more important, it is a one-step procedure, that is, zero-mean uncorrelated Gaussian processes. it bypasses the TDOA estimation step and directly estimates 3) The power spectral-densitymatricesof the sources as the sourc’e locations. This method is based onthe eigenstructure of the spectral density matrix 6f the received signals well as of the noises are known (note that this assumption implies that the number of sources d is known). and is a generalization of a method independently proposed 4) The sensor outputs are observed for acommon time interbySchmidt [14] and Bienvenu [3] (see also [16], [4], [5]) val of T seconds, whichis long compared to any of the random fortheestimation of thedirection-of-arrivalofmultiple narrow-bandsources. The method yields asymptotically un- process correlation times andalso long compared to thepropagation time of the source wavefront across the array. biased estimatesofthelocationandthespectraldensity Since the observation interval (-T/2, T/2) is finite, we can matrixofthesources,butotherwiseisnotknown to be choose to represent the received signals by either aFourier optimal under any established criterion. series The firstapplicationofa well definedstatisticalcriterion to themultiplesourceproblem was presentedbySchweppe 1 [ 151. Using a weighted least-squares (LS) criterion, which is ri(t) = ~ 1 / 2 Ri(wn)e j w n t -T/2 < t < T/2 (2) n=-m closely related to the ML criterion, he derived theoptimal estimatorand gave apartialdescriptionofitsstructure. (which implies the periodicity of Ti(.) outside the observation Recently Owsley andSwope [12] derived the ML processor interval)or by the inverse Fouriertransform of asampling for the special case of multipath propagation of a narrow-band series source. These optimal processors, like the optimal processors t for a single source, require a priori knowledge of the spectral sin (w, - - nn) 2 density matrices ofthe signals and of thenoises. t In this paper n=-m - - we present a detailed derivation ofthe ML pro(w, z sources multiple general cesser the for case. Unlike Schweppe’s, \ L - nn) /
I-
2
~
~~~
IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING,
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VOL. ASSP-31, NO. 5, OCTOBER 1983
Note that A(w,) is an m X d matrix whose ith column A i ( o , ) is a function of the location of the ith source. We will refer to it as the locationvector of the ith source, Equation (6) isvalid for every w, E B. Thus, assuming that B contains N “frequencies” {al,* . . , w ~ it }follows that we can rewrite (6) as
R=AStN
Y
Y
r
Y
r
R
= [ R ( o l ). . . R(wN)]
. . .S(UN)l
N T = [N(Ol) Fig. 1. Array geometry.
lillTI2 1
TI2
(4)
.rj(t) e j w n t dt
where
Theparticularformofnormalization(1/T1”) used here is for convenience in working with the covariance of the Fourier coefficients. Note that since T i ( . ) is real it follows that
Ri(o-,) = RT(o,)
d
~ ~ ( w =, )
aik ~
~ ( 0 e-iwnTik , )
+N~(W,)
k= 1
i = l ; . - , m (5) or rewritten in matrix notation
R(%) = A ( w , ) S ( w , )
f
Nw,)
(64
where
R T(o,> = [R1(%)
. . . R,(w,)l
ST(%z) =
[sl(%)’
NT(o,> =
Vl(0,)
‘ ‘
*
Sd(wn)l
. . N m (w,>l
-
*
(7b)
’N(cdN)]
and A is a block-diagonal matrix with entries{A(oi)}. Multiplying (7a) byits conjugate transposeandtaking expectations, using the assumption that {si(-)} and {ni(.)} are uncorrelated and zero-mean, we obtain
E [ R R t ] = A E [A SS t t]
t E[NNt]
(8)
where -t denotes the complex conjugate transpose andE [ . ] denotes expectation. Now, recalling the assumption that the observation time is mush longer than any of the correlation times of the random processes involved, it follows that the Fouriercoefficients (see, eg., [19, pp. 80-811 ) corresponding to different “frequencies” are uncorrelated, namely
EM Rw ( 4n+) 1
= K(%)
6nE
E[S(w,) S(wzP1 = P(w,> 6,1 EW(w,) N ( ~ z )=~QI( w d z6,
(94 (9b) (9 c)
where 6,, denotes the Kronecker delta and K(w,), P(w,), and
for all n
where * denotes the complex conjugate, so that we need only to consider “positive frequencies,” i.e., a, with n > 0. Taking the Fourier coefficients of both sides of (I), assuming { T k i } 6pq (37a) very low SNR, the errors in the estimation of the different sources are uncowelated. In other words, there is no degradaand also, due to thelow SNR,K(w,) = Q(w,), so that tion in the location estimate of one source due to the simulCpq ( a n ) = Q;'(an) 6pq. taneous presence of the other sources. This result is in accordance with the structure of the ML processor for this special Substituting (31) into (36) yields case, as derived and examined in theprevious section. I Jprqs(*) = 2
Re
1
6,
N
m
2 n=1 i = l
a iarp aip
v. COMPUTATION OF THE ML ESTIMATOR
asq aiq
i# s
* P p ( 4 P q ( 4QT'(wn) QF'(wn> N
. e-iWn(rip-riq+7rq-rrprp) -
w2n aip aJP a~q arq
n= 1
.pp(wn>Pp(wn> Qi'(wn)bF'(an)
. e-iwn (rsp -7sq +rrq q r p )
I
.
The structure of the ML processor described in Section I11 was presented to allow physical interpretation of the estimator. The actualcomputation of the ML estimator is done more efficiently by other methods. The maximization problem posed by the ML criterion, namely max L(R IO) 0
(38)
/
Now, since the observation time was .assumed to be long, it follows that the frequency spacing 2n/T issmall so that we can approximate the .sum over the different frequencies by an integral. Then (38) can be rewritten as
is a standard nonlinear optimizationproblem.Thus, all the known'iterative algorithms for solving such problems (see, e.g., [ O O ] ) can in principle be applied to this case. The simplest and probably the best knownoptimization method is that of steepest descent, defined by the iterative algorithm
WAX AND KAILATH: ,OPTIMUM LOCALIZATION OF MULTIPLE SOURCES
defined by the iterative algorithm
A less known method, although popular in statistics for solving ML problems, is called the scoring method. In this method the Hessian H k is,not updated and inverted at every iteration, but instead is replaced .by its expected value, whichisthe Fisherinformationmatrix,. The scoring method is hence defined by the iterative algorithm A
@k+l
is missing, simplybyparametrizingthesespectraldensity matrices and extending the search over all possible degrees of freedom, it is clear that this approach ,is computationally infeasible. A better approach is to’use the ML processor only as a“‘fine-tune”stepfollowing,forexample,thealgorithm proposed by Wax et al. [ 18J for the estimation of the source locations as well as the spectral-density matrices of the sources and the noises. ACKNOWLEDGMENT The authors wish to thank thereferees for helpful comments and the references to Schweppe [ 151 . REFERENCES
A
= @k -k J-’ g k
where J is the Fisher information matrix. The expressionsfor‘thegradient, Hessian, andFisherinformation matrix needed for carrying out these three methods are easily obtained from the results of Section IV. The three algorithmsdiffer in their convergence properties. While for the steepest descent method convergence is always guaranteed, althrough at a slow rate, the reverse is true for the NewtonRaphson method;. there, convergence is not guaranteed, but when it does occur the convergence rate is hgh. Comparing theNewton-Raphson method to the scoringmethod, it is clear that the Newton-Raphson method is more sensitive to the data, whch can be a disadvantage if the initial estimate is not close to the maximum value. Thus,the scoring method may do better than ,the Newton-Raphson method when, the initial estimate is not close to the maximum, but may not do as well near the maximum. VI. CONCLUDING REMARKS The ML processorforestimatingthelocationofmultiple sources by a passive array, for the case where the observation time is much longer than the process correlation times, was derived and its performance analyzed. It was shown that the ML processor consists of a bank of beam-formers, each focused to a different source, followedby a variable matrix-filter controlled by the assumed location of the sources. T h s processor simplifies considerably in the case that the sources are uncorrelated and the SNR is very low. In this case, the ML processor was shown to be the aggregate of ML processorsfor a single sourcewhereeach processor is matched to a different source. The CRLB onthe accuracyofthe sources’ locationestimator was also derived. The analysis of this boundforthe special use of uncorrelated sources and very low SNR showed that there is no degradation in the accuracy of the ML source lqcation estimator due to the simultaneous presenceof other sources, as could be anticipated from the structure of the ML processor for this case., We should emphasize that the ML estimator and.the CRLB were derived forthe case thatthe spectraldensitymatrices ofthesources as well as of the noises are known. This assumption may often not hold. Although one could formally construct the ML processor for the case that this information
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W. J. Bangs, “Array processing with generalized beamformers,” Ph.D. dissertation, Yale Univ., New Haven, CT, 19-71. W. J. Bangs and P. M. Schultheiss,“Space-time processing for optimalparameterestimation,” in .Signal Processing, J.W..R. Griffiths, P. L;. Stocklin, and C . Van Schooneveld, Eds. New York: Academic, 1973, pp. 577-590. G. Bienvenu, “Influence of the spatialcoherence of the background noises oh high resolut‘ion passive methods,’’ in Prac. IEEE Znt:Conf. Acoust., Speech, Signal Processing, Washington, DC, Apr. 1978, pp.306-303. G . Bienvenu and L. Kopp,“Adaptivity to background noise spatial coherence for high resolution passive methods,” in Proc. ZEEE Int. ‘Con& Acoust., Speech, Signal Processing, Denver, CO, Apr. 1980, pp. 307-310. , “Source power estimationmethod‘associatedwith high resolution bearing estimator,” in Proc. ZEEEZnt. Conf. Acoust., Speech, SignalProcessing, Atlanta, GA, Apr. 1981, pp. 153-156. G . C. Carter, Special Issue on Time-Delay Estimation, ZEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, June 1981. W. R. Hahn and S. A. Tfetter, “Optimum processing for delayvector estimation in passive signal array,” ZEEE Trans. Znform. Theory, vol. IT-19, pp. 608-614, May 1973. P. Henrici, Applied and Computational Complex Analysis, Vol. ZZ. New Yorki Wiley, 1977. C. H. Knappand G . C. Carter,“The generalized correlation method for’ estimation of timedelay,” ZEEE Trans. Acoust, Aug. Speech, Signal Processing, vol.ASSP-24, pp.320-327, . . . 1976. M. Morf et al., “Investigation of new algorithms for locating and identifying spatially distributed sources and receivers,” DARPA, Tech. Rep. M355-1,1981. A. Nehoraiand M. Morf, “Estimation of timedifference of arrival formultiple ARMA sources by poledecomposition,” inProc. ZEEE CDC, Orlando, FL, Dec. 1982, pp. 1000-1002. N. Owsley and G. Swope, “Time delay estimation in a sensor array,” ZEEE Dam. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 519-523;June 1981. B. Porat and B. Friedlander, “Estimation of spatial and spectral parametersofmultiplesources,” ZEEE Trans. Inform. Theory, V O ~ .IT-29, pp. 412-425, &983. R. 0. Schmidt, “Multiple emitter location .and signal parameters estimation,” in Proc. RADC SpectrumEstimationWorkshop, Rome Air Develop. Center, Griffiss AFB, NY,1979, pp. 243-258. F. C. Schweppe, “Sensor array data processing for multiple signal sources,” ZEEE Trans. Inform. Theory, vol. IT-14, pp. 294305, Feb. 1968. R. 0. Schmidt, “A signal subspace approach to multiple emitter locationandspectral,estimation,” Ph.D. dissertation,Stanford Univ., Stanford, CA 1981. H,. L. Van Trees, Detection, Estimation and Modulation Theory, Vol. Z.> New York: Wiley, 1968. M. Wax, T. J. Shan, and T. Kailath,“Locationandspectral density estimation of multiple .sources,” in Proc. 16th Asilomar CoaJ Circuits,Syst.,Comput.,Monterey,CA,1982,pp.322-326. A.D. Whalen, Detection of Signals in Noise. New York: Academic, 1971.
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